completeness theorem for first-order logic

E100620

mathematical theorem metalogical theorem result in mathematical logic

The completeness theorem for first-order logic is a fundamental result in mathematical logic, proved by Kurt Gödel, which states that every logically valid first-order formula is provable from the axioms of first-order logic.

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All labels observed (3)

Label	Occurrences
Gödel's completeness theorem	2
compactness theorem	1
completeness theorem for first-order logic canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T839945 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: completeness theorem for first-order logic
Context triple: [Kurt Gödel, notableWork, completeness theorem for first-order logic]

A. Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
B. Gödel's incompleteness theorems
Gödel's incompleteness theorems are two fundamental results in mathematical logic showing that any sufficiently powerful, consistent formal system cannot prove all true statements about arithmetic, and cannot prove its own consistency.
C. Hilbert’s program
Hilbert’s program was an influential early-20th-century initiative in the foundations of mathematics that sought to formalize all of mathematics and prove its consistency using finitistic methods.
D. Kripke fixed-point theory of truth
The Kripke fixed-point theory of truth is a semantic framework developed by Saul Kripke that uses partial truth predicates and fixed points to consistently handle self-referential sentences and semantic paradoxes like the liar paradox.
E. Lectures on the Logic of Arithmetic
Lectures on the Logic of Arithmetic is an educational work by Mary Everest Boole that explores the foundations and teaching of arithmetic through logical and psychological principles.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: completeness theorem for first-order logic
Target entity description: The completeness theorem for first-order logic is a fundamental result in mathematical logic, proved by Kurt Gödel, which states that every logically valid first-order formula is provable from the axioms of first-order logic.

A. Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
B. Gödel's incompleteness theorems
Gödel's incompleteness theorems are two fundamental results in mathematical logic showing that any sufficiently powerful, consistent formal system cannot prove all true statements about arithmetic, and cannot prove its own consistency.
C. Hilbert’s program
Hilbert’s program was an influential early-20th-century initiative in the foundations of mathematics that sought to formalize all of mathematics and prove its consistency using finitistic methods.
D. Kripke fixed-point theory of truth
The Kripke fixed-point theory of truth is a semantic framework developed by Saul Kripke that uses partial truth predicates and fixed points to consistently handle self-referential sentences and semantic paradoxes like the liar paradox.
E. Lectures on the Logic of Arithmetic
Lectures on the Logic of Arithmetic is an educational work by Mary Everest Boole that explores the foundations and teaching of arithmetic through logical and psychological principles.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ metalogical theorem ⓘ result in mathematical logic ⓘ
alsoKnownAs	completeness theorem for first-order logic ⓘ surface form: Gödel's completeness theorem
appliesTo	classical first-order logic ⓘ first-order logic ⓘ
assumes	standard Tarskian semantics for first-order logic ⓘ
author	Kurt Gödel ⓘ
clarifies	limits of first-order formalization ⓘ
concerns	relationship between semantic consequence and syntactic provability ⓘ
contrastsWith	Gödel's incompleteness theorems ⓘ surface form: incompleteness theorems
dependsOn	soundness of the deductive system ⓘ
doesNotApplyTo	full second-order logic ⓘ
equivalentlyStates	if a formula is not provable then there exists a model in which the formula is false ⓘ semantic consequence coincides with syntactic derivability for first-order logic ⓘ
field	mathematical logic ⓘ model theory ⓘ proof theory ⓘ
historicalContext	Hilbert’s program ⓘ surface form: Hilbert program
implies	Löwenheim–Skolem theorem (via additional arguments) ⓘ compactness theorem for first-order logic ⓘ sound and complete proof systems exist for first-order logic ⓘ
influenced	development of modern logic ⓘ formal semantics in logic ⓘ logical foundations of computer science ⓘ
isCornerstoneOf	axiomatic method in mathematics ⓘ model theory ⓘ
laterProofMethod	Henkin construction ⓘ canonical model construction ⓘ sequent calculus proofs ⓘ tableau methods ⓘ
originalProofMethod	reduction to Skolem normal form and construction of a countable model ⓘ
presentedAt	Königsberg conference 1930 ⓘ
relatesConcept	deductive systems ⓘ logical validity ⓘ models ⓘ provability ⓘ
requires	effective deductive calculus for first-order logic ⓘ
shows	all first-order validities are recursively enumerable ⓘ
states	every logically valid first-order formula is provable from the axioms of first-order logic ⓘ if a formula is true in every model then it has a formal proof from the axioms and rules of first-order logic ⓘ
usedIn	automated theorem proving ⓘ formal verification ⓘ foundations of mathematics ⓘ
yearProved	1929 ⓘ
yearPublished	1930 ⓘ

How these facts were elicited

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Kurt Gödel → notableWork → completeness theorem for first-order logic ⓘ

completeness theorem for first-order logic → alsoKnownAs → completeness theorem for first-order logic ⓘ

this entity surface form: Gödel's completeness theorem

Herbrand's theorem → relatedTo → completeness theorem for first-order logic ⓘ

this entity surface form: compactness theorem